Time, Speed and Distance Question 2 - CAT 2020
In a car race, car A beats car B by 45 km, car B beats car C by 50 km, and car A beats car C by 90 km. The distance (in km) over which the race has been conducted is:
Step-by-Step Explanation
Let the total length of the race track be m km. In a race, the ratio of the speeds of the runners is equal to the ratio of the distances they cover in the same time frame.
A beats B by 45 km:
vᴬ/vᴮ = m/(m − 45)
B beats C by 50 km:
vᴮ/vᶜ = m/(m − 50)
A beats C by 90 km:
vᴬ/vᶜ = m/(m − 90)
Since:
vᴬ/vᶜ = (vᴬ/vᴮ) × (vᴮ/vᶜ)
we chain the ratios together:
m/(m − 90) = [m/(m − 45)] × [m/(m − 50)]
1/(m − 90) = m/[(m − 45)(m − 50)]
(m − 45)(m − 50) = m(m − 90)
m² − 95m + 2250 = m² − 90m
2250 = 5m ⟹ m = 450 km
Correct Option: B
💡 FundaMakers Tip:
Look at the race positions right when A cross the finish line. At that exact moment, B has 45 km left to finish, and C has 90 km left. The gap between B and C is 90 − 45 = 45 km.
By the time B runs his remaining 45 km to cross the finish line, C has only moved far enough for the final gap to grow to 50 km. This means B runs 45 km in the exact same time it takes C to run 45 − 5 = 40 km.
This reveals their speed ratio instantly:
vᴮ/vᶜ = 45/40 = 9/8
Since vᴮ/vᶜ = m/(m − 50), setting:
m/(m − 50) = 9/8
gives:
8m = 9m − 450 ⟹ m = 450
in one quick step.